# Sum of minimum number of vertices

**Geometry**Level 5

At \(t = 0\), I begin truncating a cube. At \(t = 1\), I obtain its dual, the octahedron.

Find

\[\sum_{S \in 2^{\lbrace 0,1 \rbrace}} \min_{t \in (0,1) \cup S} V(t)\]

where \(V(t)\) is the number of vertices of the truncated cube at time \(t\) and \(2^{A}\) is the power set of \(A\).